A vector space is a mathematical structure that consists of a set of objects (vectors) and two operations (vector addition and scalar multiplication) that satisfy certain properties. These properties include closure, commutativity, associativity, existence of an identity element, and existence of inverse elements.
The basic properties of vector spaces include closure, commutativity, associativity, existence of an identity element, and existence of inverse elements. Additionally, vector spaces must also have a zero vector, distributivity, and scalar multiplication by 1.
A basis is a set of linearly independent vectors that span the entire vector space. This means that any vector in the space can be written as a unique linear combination of the basis vectors.
Linear transformations are functions that map one vector space to another, while preserving the vector space structure. This means that the operations of vector addition and scalar multiplication are maintained under the transformation.
Vector spaces have many practical applications, such as in computer graphics, machine learning, and physics. They are used to represent and manipulate geometric objects, analyze data, and solve systems of linear equations.